1. Sometimes it's simply a misadjusted float level, and the adjust
screw (or bend-the-float-arm method) work.
2. Sometimes a new chain is a little longer, and when at rest, part of
it is getting sucked into the water exit, keeping the flapper open, and
water continues to exit (and thus trigger the fill valve.) Check that.
3. Sometimes the floats get corrupted, perhaps from using chlorine
tablets which eat away at their plastic or rubber. You can buy a
replacement float to fix that.
4. If none of the above help, then it likely is the fill valve
assembly, the vertical structure that looks like an airport conning
tower. You can still buy the older brass ones, or you can switch to
the plastic ones. BUT BEWARE; particularly if you have hard water like
I do, the Fluidmaster plastic ones fail every 6 or 7 years- PITA. The
replacement of the fill assmebly is annoying or worse (but DIY saves
yourself $150 from the plumber: read the instructions, turn off water
valve, drain the tank, drain the tank MORE, borrow your wife's turkey
baster to DRAIN MORE, loosen the appropriate attachment fittings at the
water inlet, and where valve seats on ceramic toilet, remove, watch
water you DIDN'T drain flow over the floor, install new valve, being
careful to use the right washers depending on your water inlet type.
The Fluidmaster instructions are quite good.
5. The self-cleaning Chlorine Tablets are enemies of these rubber and
plastic systems; avoid.